Question

The angles A, B, C of triangle ABC, are in AP. If b: c = \(\sqrt{3}: \sqrt{2}\), then angle A is (a) \(60^{\circ}\) (b) \(45^{\circ}\) (c) \(75^{\circ}\) (d) \(30^{\circ}\)

Short answer

Question: In triangle ABC, angles A, B, and C are in arithmetic progression, and the ratios of sides b and c are as \(\sqrt{3}: \sqrt{2}\). Find the value of angle A. Answer: \(30^{\circ}\)

Step-by-step solution

Recall the Arithmetic Progression Definition

In an Arithmetic Progression (AP), any term can be expressed as the sum of the first term and the product of the common difference and the term's position in the sequence minus one. So in this case, if A is the first term, B is the second term, and C is the third term in the AP, we have: B = A + d C = A + 2d This shows B and C in terms of A and the common difference (d).

Apply Angle-Side Relationship

In a triangle, the angle opposite to a larger side is larger. As we are given b: c = \(\sqrt{3}: \sqrt{2}\), and since \(\sqrt{3} > \sqrt{2}\), angle B is greater than angle A. In the same manner, angle C is greater than angle B. So, we can infer that A < B < C.

Express the Conditions in AP Terms

Considering the relationship A < B < C, we can say that A should be the first term in the AP, and based on the information from step 1, the angles A, B, and C can be expressed as: A = A B = A + d C = A + 2d

Use the Property of Triangle Angle Sum

In a triangle, the sum of the three angles is 180 degrees. Apply this property for triangle ABC : A + (A + d) + (A + 2d) = 180

Solve for Angle A

Simplify the equation from step 4 and solve for angle A: 3A + 3d = 180 A + d = 60 Since A < B < C and B = A + d, angle A should be the smallest angle in the triangle. So, A = 30 degrees. Hence, the correct answer is (d) \(30^{\circ}\).

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. For example, the sequence 2, 4, 6, 8 has a common difference of 2. When the angles of a triangle are in AP, they form an ordered set where each angle differs from the others by the same constant value.

In the given problem, the angles A, B, and C of triangle ABC are in AP. If angle A is the first term, then angle B is one common difference away from A, and angle C is two common differences away. This characteristic of AP helps in simplifying mathematical problems involving sequences because it creates a predictive pattern which can be used to establish relationships between terms.

Triangle Angle Sum

The triangle angle sum property is a fundamental characteristic of Euclidean geometry; it states that the sum of the internal angles of any triangle is always equal to 180 degrees. This property helps in deducing unknown angles or verifying the accuracy of geometric constructions involving triangles. It is crucial when solving problems related to polygonal shapes and is often used in congruence and similarity tests for triangles.

When dealing with problems like the one in the exercise, the triangle angle sum property allows us to set up an equation involving all the angles. We have the angles A, B, and C for which we apply the sum property as A + B + C = 180 degrees, which then becomes a tool to find missing angles when one or two angles are already known or when there's a relationship established between the angles—such as being in arithmetic progression.

Angle-Side Relationship in Triangles

The angle-side relationship in triangles is another pivotal concept in geometry which specifies that the size of an angle in a triangle is directly related to the length of the opposite side. Accordingly, the larger the angle, the longer the opposite side and vice versa. This principle is key when performing a variety of tasks like determining whether a triangle is scalene, isosceles, or equilateral, as well as when applying the Law of Sines and the Law of Cosines in trigonometry.

In our given exercise, the relationship b: c = \(\frac{\sqrt{3}}{\sqrt{2}}\) suggests that side b is longer than side c because \(\sqrt{3} > \sqrt{2}\). Since the angle opposite the longer side must be larger, we deduce that angle B is larger than angle C. Hence, we can see how the knowledge of angle-side relationships complements our understanding of angle sequences and assists in solving for unknown angles.